I'm currently very slowly making my way through Geometric Algebra for Physicists by Doran and Lasenby. The book is a delight to read, but I'm not a mathematician, and this article is showing me that my small amount of understanding is...not nearly as deep, and especially not nearly as rigorous, as I would like. I should try to re-read with Eric's criticisms in mind.
For a physicist it eventually becomes necessary to understand exterior algebra.
This is often done in the context of differential forms, but of course can be brought back to vectors easily. With those well established tools GA doesn't offer much. This blog post seems to point out exactly this fact.
I always come back to Schutz's Geometrical Methods of Mathematical Physics as my reference for notation, but I agree. I came to this by way of General Relativity, so that colors my perceptions. The few treatments of GA that I've looked at (briefly) weren't very clear about the distinction between 1-forms and 1-vectors and seemed to assume Euclidean metric everywhere, so I left thinking that it seemed a little weird and not quite trusting it.
In any case, my experience is that the coordinate-free manipulations only go so far, but that you pretty quickly need to drop to some coordinates to actually get work done. d*F=J is nice and all, but it won't calculate your fields for you.
Note don't to be confused with Algebraic Geometry they are different;
When I first came across this topic it was eye opening, especially the fact that you could squeeze Maxwell's equations into one and the fact that pseudovector create by cross product from physics is just a bivector which in 3d could be represented like a vector orthogonal to the plane created by the two vectors in the product
Hah! I knew Sudgy would be your first link. There's relatively little content about GA, but once you're hooked you begin to consume it all.
It's like you never got to study Relativity, conformal geometry, electron spins, quaternions, etc then someone comes with a simple cheat code which introduces you to these topics gently. It's like what category theory wants to do for mathematics, but simple.
A physics engine in 100 lines (Gravity, Hook, and damping laws are just one line each!), and you can go from 2D to 3D to 4D by changing a single parameter: https://enki.ws/ganja.js/examples/pga_dyn.html
Note though that GA is extremely niche and offers very little beyond exterior algebra. Exterior algebra in the context of differential forms is standard part of the postgrad (and sometimes undergrad) Physics curriculum.
A lot of the (valid) criticism can be summarized by the "No scalar product necessary. Get rid of it" point (early in one of the tables). The next point on inner product is good, and related.
The point is: there is no natural or distinguished isomorphism from a finite vector space to the vector space of linear functionals over it. We know they are isomorphic- but there are many such isomorphisms with little to distinguish between them. The note's "arbitrary multivector, G" fix for inner products is good. The idea is most of us are using the same G that looks a lot like an identity matrix. And as long as you always use the same G you uniquely identify the inner product by equality (as the note does). But if you accidentally combine work that is using different G you run into trouble. You can end up with different Gs if you don't worry enough about naming your basis.
A lot of this keeps getting re-invented as variations of tensors, the exterior algebra, wedge product and so on. I think most of these end up being more bookkeeping than consumers want.
Not a physicist, but this seems to depend on the physical setting? Seems there are usually metric with physical meaning in continuum mechanics, e.g., elasticity or GR, but not so much if one is working in say geometric mechanics — one can define a Hamiltonian flow on a symplectic manifold without a metric.
As a fellow game dev this article should be targeted at readers like me. But my eyes can't help but *immediately* glaze over as soon as I read all the math notation.
I'm pretty ok at 3d video game math. I do lots of work with matrices, quaternions, vectors, and friends. It's not particularly difficult.
I can't for the life of me read mathy math. Wikipedia Math is inscrutable hieroglyphics. It's quite frustrating.
I wish someone would write a "foundations" article or book that spoke in language understandable by normal humans. Or at minimum had a bloody legend that explained what all the %&(#%& symbols meant.
"It just means that there’s a lot of low-quality stuff under the same label, which has made that label questionable, and if you want to sift through it you have to be ready to filter for quality yourself... At some level GA is trying to “democratize” geometry."
Hit me just now reading the case against GA linked in this thread, maybe the same way Munger remarks on Costco's counter-intuitive membership fee, the notation and systems of "real" math as we encounter them like you say in Wikipedia is the field's membership fee keeping out the riff-raff from overwhelming the gathering place and making the entire store burn down.
"But the point is to make the existing math more intuitive, not to discover new results. The fact that research mathematics is generally not concerned with making calculation and intuition easier to think about is, I think, a giant failure that it will eventually regret. There’s as much value in making things easy to use as there is in discovering them."
Honestly I'm surprised no one has done the programming/CS version of it where all the symbols and formula are explained with code. I'm rusty as hell on math but seems like it should be pretty reasonable to translate a lot of them to functions or similar per symbol.
Imagine the idea being used for more complex math operations. Summation/Product notation is pretty straight forward (if you were ever taught it). The vast majority of math notation is not. Or rather most people are not taught most math notation. And even if you were taught it it's really hard to remember!
As a former mathematician from one of the more elitist subfields of math, I have to say this is the first time I’ve heard anyone claim geometric algebra lacks rigor.
The subject doesn't lack rigor, but all of the popular and quite a few of the mathematically minded introductions to it do. Bourbaki and Chevalley did a good job introducing Clifford algebras properly some 80 years ago, but even this has been forgotten by the modern bloggy expositors.
Unfortunately, the Lundholm and Svensson text you've cited suffers from the same problems I wrote about in my post. Definition 2.7 connects the interior products to the scalar product, not the inner product. Definition 2.8 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras.
It fixes the worst problem, which is the well-definedness of the interior product, since it defines the product explicitly on the standard basis rather than through a bunch of not-obviously-consistent axioms. It is too basis-dependent for my tastes (it should depend on the symmetric bilinear form, not on the basis). As for the sign conventions, I think they're ultimately matters of taste, and while I agree with you about the inner/scalar product, I disagree about the derivation-ness of a contraction (I like it to satisfy the Koszul sign rule, which allows signs to be inserted only when some factors move past each other).
My ideal approach is along the lines of Chevalley's "The Algebraic Theory of Spinors and Clifford Algebras", Chapter III, but he doesn't get to much geometric algebra.
> even by the 90s/00s, GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots
> those reasons disproportionately attract people who are not actually capable of rigorous mathematics, or are slightly prone to conspiratorial thinking, or are otherwise slightly deranged
> if you look around for papers that explicitly talk GA, they are very disproportionately (a) non-theoretical, (b) poorly-written, (c) trivial, i.e. restating widely-known results as if they’re novel, (d) only citing other GA papers, and of course (e) just plain crackpotty
A few Reddit comments appear to agree at that non-mathematicians (e.g. amateurs and game programmers) use geometric algebra non-rigorously.
I suspect the issue is not about mathematicians who know how to be rigorous. Both posts are complaining about how mathematical outsiders are using and teaching geometric algebra non-rigorously.
Well the real issue is that the GA proponents do this weird bait-and-switch where they talk about caring a lot about being intuitive and simple, and then they do a bunch of nonsense with the geometric product that makes no sense. It's all at least possibly rigorous (depends on who you're reading), but the thing I want people to question is: is it good? Is the GA way better?
When GA has clean formulas for calculating stuff, sure. But should you write all your linear algebra formulas in terms of the geometric product? heck no. For most people GA is their first exposure to wedge products, though, and those actually are great, so they think that's GA. No, that's just ordinary well-known material that should be in linear algebra classes already. What GA adds on top of that is mostly really weird, although there is a kernel of quality inside it somewhere.
What's it even mean to use Imaginary Numbers or Quaternions "rigorously"? It's Geometry. The point is points, lines, applying transformations. If anything the field needs less rigor and more friendly intuition.
Right, complex numbers and the delta function lacked rigour for years and physicists and engineers didn't care because it turned out it captured the logic sufficiently well that useful stuff could be gained with it. Then the mathematicians did the rigour bit and everyone was happy again.
yeah, oops, that was supposed to be tongue-in-cheek. No, I'm just a guy who likes math, and I try to write things that make seem correct. I just don't want anyone, like, citing me on something... like, check the work yourself if you want to use it, I'm not a reliable source.
Anyway, go read a bunch of GA books and papers and you'll see exactly what I'm talking about there. At some point in the past (like ~8 years ago) I think I had read or at least skimmed everything that had ever been published on the subject. And some of it's good! Although at times, like, unnecessary. The rest, though... yikes.
Geometric algebra was, for me, an easier on-ramp to more advanced topics in mathematics, leveraging the geometric intuition built into these objects. I learned vector algebra in university, so extending the idea of a direction with magnitude and orientation into areas and volumes with direction and magnitude was a manageable step. Projective algebras similarly allow the construction of points, lines, and planes with orientation and magnitude. Learning that you can generate transformations by dividing one object by another was also really neat. Then you can take the logarithm of that transformation to perform linear interpolation. Geometric algebra is a tool that I use to manipulate geometric objects as easily as I would manipulate real numbers.
The hard part was getting there amidst a host of confusing and conflicting source materials, as this post highlights, and as its author helps proliferate. I used Eric Lengyel's materials a lot in my journey, but I really dislike his decision to represent points as vectors. Notice how every transformation in his poster [1] uses the antigeometric product and antireverse? These operations are only necessary because he's defined everything in the point-based dual space and has to bring everything back to the plane-based space to perform transformations. But he has a wiki and posters, and I'm just here doing my own thing, so I guess that's that.
Alan MacDonald's two books [2] were great, and I would recommend them as an introduction to geometric algebra. Work through the examples and you will learn the material.
Squaring and Cubing should be geometric ideas - not just another way to do 1D multiplication. We need new operators for transforming a number into a higher dimention.
As mentioned elsewhere, that's the wedge product of exterior algebra. Standard stuff first introduced in 1844 that has been foundational to a ton of modern math.
The claim that geometric algebra is not rigorous made me confused. But the lack of proper references is much more confusing to me. Back in the days when I was doing research in Chevalley groups the definitive and most cited source was (and still is) Geometric algebra by E. Artin. And there are some more recent treatments recognized in math community, e.g. classical groups and geometric algebra by L. Groove and the geometry of the classical groups by D. Taylor. At least this books made everything rigorous and clear. But may be there is some other field with the same title?
I've heard about book by Doran and always thought that it's more about applications.
They are talking about different stuff. Geometric algebra is here the name used for the popularisation of a brand of Clifford algebra. The popularisation books are often written by physicists and engineers, and their maths is shaky, because they care more about applications than about proofs and rigour.
There are exceptions, for example I like the two books by Alan Macdonald about the topic:
I think Macdonald's book is very concise and clean compared to others, but it does have some of the same issues that I wrote about in my post. In particular, Definition 6.15 gives one of the problematic definitions of the inner product, and Definition 6.23 gives the same broken definition of dual that has an inconsistent orientation and fails to extend to the degenerate metrics of projective algebras. Has also says, at the bottom of page 111, that the Hodge dual can't be defined directly in the exterior algebra, which is not correct.
> Virtually all authors define the dual of A […] and there seems to be no consensus as to which one is best. It doesn’t matter because every single one of these definitions is crap.
I always had the feeling that the bivector.net guys are using a bit weird definitions for certain operations, which then result in weird formulas further down the line. Nice to see Eric putting in the effort to examine their mistakes.
Where are the mathematicians who could be describing Geometric Algebra correctly?
For a professional working in linear algebra, supervising a graduate student, it should be straightforward to translate these "higher order" linear algebra objects into the geometric algebra model, without publishing technically incoherent mistakes.
"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.
Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.
How is complex / imaginary numbers not what they are doing? Numbers that square to -1, 0, and 1 are the bread and butter of the GA I know. Exploring different combinations of types of imaginary numbers and their products and space describing algebras. (including naturally quaternions, duel quaternions, i-rotation, nilpotent, ect)
Typically GA people are working with real algebras, meaning the coefficients are real, and things like a square root of -1 appear as some object in the algebra (like a 2-blade). But you could also have a Clifford algebra with coefficients in e.g. the complex numbers or fields of finite characteristic.
In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebras:
As this shows, ei and ej are imaginary numbers. This confusion is the biggest issue in GA to me. Typically Linear Algebra users don't use imaginary numbers either. It's all connected when doing geometry though and its pointless to draw invisible lines between these concepts.
We have no way to talk about more general types of complex / imaginary numbers besides rigid math lingo that provides no geometric intuition or grace for geometric imagination.
You seem to be calling complex numbers imaginary numbers, but they're not the same thing. Imaginary numbers are a subset of the complex numbers consisting of the imaginary axis without 0, e.g. i, 2i, -3.1i. Complex numbers also include the real numbers and all combinations of real and imaginary.
I'm glad you seem to know what I mean. I love imaginary numbers like the Square Root of -1 or the Square Root of 0. They can only be used like:
nil²=0
i²=-1
j²=-1
And combining them into complex forms like:
(i + j)
(nil + i)
What can I call this general idea of using imagination to determine new number rules and combining them together?
If I call this GA or Clifford Algebra in a math community will it trigger rigor admins to ban me from talking because I'm not using their terms?
I wish we had artistic imaginary math communities for exploring Geometry without rigor turing everything into Semantics. Geometry literally doesn't need semantics if you agree on points, lines, planes ect. Algebra to me should just be simple maps from clifford numbers to examples of intuitive geometry / physics.
People should stop calling that set "imaginary numbers". It's net bad. That set isn't really worth it to be called anything. Maybe "pure imaginary numbers". Also "imaginary numbers" is used for the complex numbers already.
People also like colloquially saying "The Square Root of -1". Technically wrong but personally meaningful.
GA is mainstream once normal people start joking about the square root of zero. We need to imagine and teach even more imaginary numbers one day.
I'm a big fan of Eric Lengyel, and I never would have ever gotten my arms around Geometric Algebra if it hadn't been for his books and articles. How many people can do theoretical math AND code a state-of-the-art game engine? The guy is a walking, coding miracle.
So if I'm a little critical of the tone of the article, it comes from a place of love. There has been a very toxic, clickish vibe in Geometric Algebra circles, which have lead to some pseudo-disputes among those who should be natural allies.
One such is that Gunn, et al, prefer to represent a 3D vector using a dual basis (e.g. [a1, a2, a3]^T = a1*e32 + a2*e31+ a3*e12) whereas Lengyel prefers to just represent them as a1*e1 + a2*e2 +a3*e3. Some really unfortunately hostile back and forth arguing about which one is "the right way"--when in reality, it's a big-endian vs little-endian thing. One of the best parts of projective geometric algebra is that you can flip back and forth to the dual representation whenever you want to, according to what makes sense to you--and what makes the problem at hand easier to solve. Moreover, if you look at the actual calculations doing it one way vs doing it the other it's the same damn exact numbers being multiplied/added in the same damn way. It's not quite as silly as arguing about what font numbers should be printed with, but it's pretty close.
The tone of the article reflects wounds which are still pretty sore from these sorts of battles. So I understand. But sometimes the invective goes a bit to far.
An example of this is his critique of Gunn's initial cut of dualizaiton for PGA. The fact that e0 is not invertible is a big, fat wart, for sure. And frankly, I spent weeks trying to understand Gunn's workaround and its wierd lingo. J-map? What in the world is a J-map? I finally understood the concept, but I've never found out what "J" stands for :-) And Lengyel's treatment is much smoother and more coherent.
Yet, I don't think Gunn should be criticized for it at all. It was an act of courage for Gunn to come up with his janky J-map, and not let its janky-ness stop him and the rest of the field from moving forward. Sometimes that is exactly what is required in mathematics. For example, infinitesimals. For centuries, mathematicians from Archimedes to Newton found them indispensible, even though they had absolutely no coherent mathematical foundation. In point of fact, if you listen to, say, a Feynman lecture, you'll find that they are still indispensible today. But they didn't have any kind of mathematical foundation until the 1960's, when Robinson found a way to coherently axiomitize them.
I saw a video of Freeman Dyson once, where he was talking about how he was able to prove something which was a longstanding open problem. He described his proof as "very ugly" and then went on to say (with tongue partly in cheek) that you can judge how great a mathematician is by how many ugly proofs he creates :-) Because the first time something is proved, the proof is almost always very ugly. It's not until other mathematicians come in and find connections with other branches of math, and start being able to come up with more elegant proofs.
So let's celebrate the ugly, messy, janky-ness which is the reality of how mathematics is actually created, and the courage of the mathematicians to not rat-hole and bike-shed.
Eric Leyngel's presentation of projective geometric algebra is, IMHO, far more coherent and elegant than any other presentation. His books (and his source code) are a joy to read. For a newb like me, it is far easier and quicker to absorb. Isn't that good enough? Did he really have to go on to flame everybody else to a crisp? sigh like I said, he has been subjected to very unfair and toxic pillorying, and the wounds are still fresh, so like I said, I understand. But its very regrettable nevertheless.
So, I've shipped a game with his engine, so have a unique relation ship with him, and he's a complicated individual.
To be clear, he is a very talented, intelligent individual. But he form strong opinions, has a very hard time taking criticism, or understanding other people's differing context, and has a hard time not taking disagreements personally.
This is really nothing new from him. I think the best way to interact with him, is hear out his points, and use them to synthesize your own viewpoint. And be careful to take his views as your own. I think if you do that, you can learn a lot from him, but to be careful as his strong disagreements are often more nuanced then he makes them out to be.
Do you have a blog post or write-up about the experience? I'd love to know more about what working with both the engine and the man is like--and I suspect there's more stories to be told :-)
Lengyel's post doesn't discuss the whole plane-based vs. point-based war at all, and he makes it clear in his book that both approaches are equivalent. He actually says both are happening at the same time whichever way you look at it, but I'm still trying to wrap my head around that. Gunn is not one of the authors of the material Lengyel is talking about in his post.
I am not sure I understand the plane-based GA model, but I imagine it's that in exterior algebra there is another product that's totally dual to the wedge product (people call it the "antiwedge product" or "regressive product"), and the plane-based and point-based models just swap the two symbols.
Hi Alex -- In PGA, every operation comes in pairs. There are two exterior products, two inner products, and two geometric products (and the list goes on). If points are represented by vectors, then the quaternion-like sandwich qpq* with the geometric product, where q is now a more general operator in the algebra, always fixes the origin. Thus, it cannot perform Euclidean isometries in regular space because those (in general) move the origin. However, a fixed origin in regular space means that the horizon is fixed in antispace, so if you were to reinterpret vectors as planes instead, then you do get the set of Euclidean isometries that you want. If you had no knowledge of the geometric antiproduct, then you would just say "vectors are planes" and call it a day. That's where plane-based GA comes from. Just use the geometric product and interpret all geometries in antispace instead of regular space. But this throws out the geometric intuition shown in Figures 2.4, 2.5, and 2.7, where vectors, bivectors, and trivectors are simply projected into the w=1 subspace to de-homogenize points, lines, and planes. Furthermore, we need the general notion of product-antiproduct pairs to get things like norms working, anyway, so we might as well use them to avoid dualizing all the geometries.
The space/antispace duality is discussed in Section 2.6, and the fact that the geometric product fixes the origin is discussed in Section 3.5.1. (In case anyone else is wondering, I know ajkjk has a copy of my book.)
I don't think Lengyel is criticizing anything for just being ugly. When he says something is janky, it's because there's something mathematically wrong/incorrect/broken with it.
> Yet, I don't think Gunn should be criticized for it at all.
The article dedicates itself to critiquing the techniques, but spends no time that I can remember talking about the human beings and their feelings. That’s how I write professionally, and I’ve found it can really upset people who infer that critique of some thing is therefore critique of some person — even when no such inference is intended by the author.
> Did he really have to go on to flame everyone else to a crisp?
No one person is flamed to a crisp here, but some people’s mathematical works are absolutely set on fire. If this had been a critique about the people in geometric algebra, I never would have read it at all, because that’s not what interests me. Critiquing cargo-curled erroneous hearsay as invalid resonates strongly, especially with the focus on the math instead of the people.
As a newb, I learned a great deal about mathematics from reading this, but I still don’t know who any of the people involved are. Isn’t that the holy grail of professional critique: it’s about the work, not the worker?
Or, am I missing something where this article is personally attacking people rather rhan people’s work output?
He repeatedly makes assertions that "the authors" have "no idea what they're talking about". The difference between flaming an individual person and "the authors" is thin at best, arguably only different in that it flames more than one person at once. And honestly, to the extent he's correct, I think that's fine. But let's not kid ourselves about what we're reading.
Well, he describes some folks' work as "crackpot-level bullshit" and repeatedly states that other authors "have no idea what they're talking about". These seem to me to be personal critiques.
Generally, I'm not sure the distinction you draw is quite so clear cut -- if I say somebody is great but all their work is hot garbage, that sounds fairly personal.
So much education content is so very poor. And even the best of it... A first-tier physics professor at a munch was delighted - he regaled believing he had found an error in a highly-regarded introductory textbook! But, upon many day's of thought, and several close reads of the text, he had realized it had been very carefully worded so as to be not incorrect. And so he was so delighted - yay! He thought of this as a good state of affairs, reflecting well on the text, and associated instruction. I... afterwards wished I'd pointed out that the target audience for an intro textbook, was perhaps not well modeled as first-tier experts with a week to wrestle with and closely ponder a paragraph in order to avoid being misled.
I'm unclear on how we get better at this. I've seen OER texts with open errata databases still struggle. Perhaps a github-like fine-grain (Xanadu-like transclusion) wikipedia? Or "nLab all the fields"? Or... ??
I tried to report an open calculus textbook from Rice University’s talking about relativistic mass as an error (It’s pretty well-established as a bad concept in physics education at this point as opposed to the momentum energy 4 vector) and they wouldn’t accept my feedback.
Yeah - I've seen "but it's on lists of most common misconceptions" closed wontfix. Errata are good for "author:oops,tnx", but work much less well for confused authors and bad calls, and not at all for judgment calls and alternate approaches. Some other mechanisms are needed.
I don't know the solution either. My stats professor religiously attended and espoused these "teaching stats" conventions. But the end result was him always deferring to how the committee did things. The entire pedagogy including how he answered questions. I really didn't like this solution and it made me hate stats until some reacquaintance with it in discrete math.
But then if you're at the mercy of a professor who does things their own way, you can have cases like you give.
One thing that helped was getting syllabi from future potential classes and comparing which textbooks they used. My advisor helped me do this and I credit it with making my senior year more tolerable.
I'm not the OP, I'm not a part of this community, and I don't know if the thing I'm about to complain about is what the author was thinking of with this comment, but as someone who was trained as a mathematician and who has read some of the popularizations of geometric algebra that sometimes get posted to HN, there is a tone that some (though probably a minority) of them take that I find pretty obnoxious.
These pieces are the ones that take the position that geometric algebra is this super secret anti-establishment mathematical samizdat that *they* don't want you to know about. They'll pit themselves against "mainstream mathematics" and say things like, "in differential geometry you do X, but you shouldn't do differential geometry; you should do geometric algebra where we do Y, which is so much better than X."
My reaction is always, "My friend, you are doing differential geometry!" Clifford algebras --- the objects that the geometric algebra people study --- are firmly within the "mainstream" of mathematics; there's simply no conflict here, at least not of the sort that these writers often seem to be imagining. It's great that people are enjoying learning about Clifford algebras. I think Clifford algebras are really fun! But we can all just come together and enjoy them together, and I think this "join me in taking down the cabal of gatekeepers who are suppressing the truth" attitude is unnecessary and turns off a lot of people who might otherwise be fun to engage with.
If you're into this stuff and feel like this doesn't describe you or the people you know, then that's great, keep doing what you're doing! But it does exist and I wish it didn't.
I used GA as a way to bootstrap into 'real' Clifford algebras, and a way to get over a "reader's block" when it came to Lie algebras, tensors, and (finally) algebraic geometry. I'm not sure GA is great math, but it was really great way to learn "advanced math concepts" for "basic..ish math". Personally, I like Alan MacDonald's GA books — they're a great way to learn more complicated concepts, but couched in a very approachable geometry/visual learning style.
That sounds like a fun and satisfying process! I realize my comment could be taken as denigrating all the people who write about this stuff, but that's certainly not my intention; I've also enjoyed a lot of the visualizations and geometric explanations that people writing under this heading have come up with. My complaint is really just about the ones who take this oppositional attitude, and a big part of why I think it's such a shame when that happens is that there really are some very cool ideas here, so it's sad to see walls being raised for no good reason.
Mathematicians will take a moment denigrate Geometric Algebra as "linear algebra with a uselessly nonstandard notation", ignoring that we should prefer a less awkward way of structuring linear algebra than "pseudoscalars" and "pseudovectors".
>
Mathematicians will take a moment denigrate Geometric Algebra as "linear algebra with a uselessly nonstandard notation", ignoring that we should prefer a less awkward way of structuring linear algebra than "pseudoscalars" and "pseudovectors".
I have never heard a mathematician using the terms "pseudoscalar" and "pseudovector". These rather seem to be common terms among physicists.
> "linear algebra with a uselessly nonstandard notation"
Let me chime in that as a physicist (who does use the "pseudo" stuff occasionally) I very much share this opinion.
The notation may be really cool and compact, but I just do not see the benefit - for example, d*F = j and dF = 0 is compact enough for me.
It is all fine if people use this language to learn linear algebra or differential geometry. And maybe it has a use for numerics or computer science. But I am quite sure that the geometric algebra formalism will not be widely adopted in physics any time soon. Sorry.
> I strongly believe that if GA would make this distinction they would lose a lot fewer people. It is a completely interesting and useful thing to talk about “a representation of a particular class of operations that makes composition and inversion easy”, and completely offputting when you blur the distinction between operators and geometric objects themselves, and write every operation in terms of the geometric product when only a few of them are really compositions of operators.
I can't tell what "a few of them" refers to. What is this potential distinction between operators and geometric objects? Sounds like the the distinction between a group action and a group object?
I am willing to believe that GA is an unnecessary renaming of other simpler things, and also that it has these kind of culty vibes, but I'm focusing on the claim that (I understood as) "unifying the operators and geometric objects" is a bad feature rather than a good feature.
So, I've amended the article some since posting it because the main objection I got from a few GA enthusiasts was this idea of the GP as being used to compose operators. Which I had barely noticed the importance of because it's so hard to identify in the texts! Although once they mentioned it I began to appreciate that, when the GP works and is useful, this is why. So I changed things to address that point more directly... but I didn't really find the energy to do a perfect job of it (like researching all the examples I would need to make the point more clearly), and it's a bit muddy at the moment. Like at the spot you pointed out. I need to figure out a clearer way to say this stuff...
What I am getting at is that if you go read, say, the Doran/Lasenby book, they start out talking about multivectors for areas and volumes and etc---and they do all this with the GP. Which makes no sense! Ever calculation they do leaves you think "huh?" The GP makes no sense at all if you're talking about units of length, area, volume, etc. Its transformation laws, its composition laws... you end up having to undo it all afterwards with a bunch of janky other operations.
But if you talk about the GP for composing reflections to make rotations, it's fine, that makes sense. I just really want this distinction to be made more clearly. I'm only interested in the GP when it corresponds to an explicit geometric operation. Nobody makes this distinction as clear as I want; I hope to eventually find a really sound version of the argument and then write it out as another article.
Roughly speaking it's equivalent to conflating the sense of a complex number as a vector with a complex number as an operator on vectors. Yes, they're isomorphic, but given a vector in R^2 there's no intrinsic sense in which you should be able to interpret it as also being the operation of multiplying by r e^(iθ) on other vectors. Pretending like they're the same thing is just bewildering: that identification between vectors and operations should be something you have to explicitly construct. For starters, if you change bases for (x,y) the vector should rotate but the rotation operation shouldn't change. That sort of thing. GA is making this same confusion but on a larger scale.
My position is that the geometric product and antiproduct are good for one thing, performing transformations with sandwich products q ⟑ p ⟑ q̃ or q ⟇ p ⟇ q̰ and composing those transformations. Literally everything else (join, meet, contraction, expansion, projection, inner product, norm, ...) can and should be done in the exterior algebra without any geometric products.
agree but I am still trying to grok the divine truth as to why exactly that is. What's up with the sandwich products? Why do they work? I guess it is like a change-of-basis for a matrix (PAP^{-1)) but I still don't quite see why, and why it works as a change-of-basis on multivectors, not just vectors.
Well by "taught in linear algebra" I mean "taught to undergraduates in their first or second year". It is totally bizarre that people learn about determinants, matrix minors, curl, magnetic fields, angular momentum, etc... but not about wedge products, in which all of these things are much easier to understand.
Totally agree! And this is why I sympathize with GA fanatics. Although it might not really make sense to do the rebranding or be maximalist about the geometric product, I do think there is a shared goal of making these things easier to understand.
Ya, as I write in my article, I'm very on board with the "change the curriculum to make more sense" project. I just want to be talking about how to do it in a critical way, rather than just going with the GA way, which requires, in my opinion, a lot of justification that I haven't seen.
You were not in the target audience. This was written for people already familiar with the theory of geometric algebras.
Like any other mathematics paper, it cannot include the content of a whole textbook inside the paper, to provide context for non-mathematicians or for mathematicians with a different specialization.
Here on HN there are every day links to "walls of text" that are trivial to understand for those familiar with computer programming, but which would be completely impossible to understand for a non-programmer.
I believe I am already familiar with the theory of geometric algebras. (Funny you mention this, since the author clearly believes that most textbook authors are not familiar with the theory of geometric algebras, and that they blindly copy each others textbooks without understanding it.)
.. how is _my_ comment flagged, when all but one other comments are just randomly jumping on the title (which has nothing to do with the article btw)? I at least claimed that I've read the first big chunk of the article (before dismissing it), while none of the other comments are about the article or show any sign that they've read the article. (And the other comment dismisses the article too, despite "being a big fan of the author".)
This is not a critic of all geometric algebra and especially not of its more basic parts. Therefore it is not an excuse for not grasping e.g. what Hestenes has written about GA, or what Eric Lengyel himself has written, e.g. in his new book that is advertised in this article.
It is a critic of many books about geometric algebra, which have made attempts to expand and further develop some parts of its theory, but those attempts have not been thought carefully and they have produced various inconsistent or useless definitions.
It is also a critic of attempts of presenting geometric algebra as preferable for applications where in fact it is not optimal, by showing misleading "benchmarks". Unfortunately this tactic is not at all specific to geometric algebra, but it is frequently encountered for almost any kind of algorithm known to mankind when it accumulates for one reason or another some kind of fan base.
Calculus had poor foundation and was thus logically incoherent from Newton/Leibniz' discovery to roughly the middle of 19th Century. None-the-less it was a powerful tool and most of the key theorems were discovered then.
The basic situation, I think, is a set of tools can be consistent in the way mathematicians use them but in the way the mathematicians explain them. And the tools can be very useful despite this.
So my guess is saying "it has poor foundations" isn't saying "I'm against it, it's worthless"
Yeah I’m pretty well-versed in free constructions of various algebraic objects and how this would interact with things like a quadratic form, etc, but couldn’t sort out GA (I think the authors of “GA4CS” had a very different sort of computer scientist in mind). When I saw the geometric algebra constructions clash with constructions I was already familiar with, I generally got suspicious and lost interest.
I’m actually quite interested in checking out Lengyel’s book. It looks rock solid.
(I did not downvote the parent of this comment, nor am I angry. I can't speak for anyone else.)
Seems a bit of a weird request, but sure. I assume the point of it is some sort of credential-test, so I'll focus on things relevant to that.
My name's Gareth McCaughan. I'm a mathematician, though I've never worked specifically in the field of geometric algebra, nor in the foundations of mathematics, and I haven't been doing proper pure-mathematics research for several decades. I have a BA degree, a master's, and a PhD in mathematics from the University of Cambridge. After that I worked for a couple of years as a junior research fellow (i.e., the lowliest type of academic staff member) at one of the colleges there, and then moved into the Real World where the mathematics is easier and the pay is better. I've worked for a number of startups doing broadly mathematical (often but not always geometrical) things, and after a chain of acquisitions I'm now working for HP.
The post you linked to claims to be philosophical as well as mathematical. I have no particular qualifications in philosophy, unless you count the irrelevant fact that PhD stands for "philosophiae doctor". (I've read a fair bit, but obviously that proves nothing much.)
Of course, whether it's true that your earlier post has nothing to do with geometric algebra doesn't depend on my qualifications or abilities or knowledge or whatever; it depends on what that post is about and what geometric algebra is about. So I'll say a few words about those.
Geometric algebra is a quite specific thing in mathematics, related to differential geometry and exterior algebra and the like. It involves a construction in which one starts with a vector space (say, R^3, which one can think of as the place where ordinary three-dimensional things live), and embeds this in a larger structure called a Clifford algebra whose elements can be multiplied together in a meaningful way. This turns out to be a useful framework for various things in physics and computer graphics.
The linked post is titled "An exploration of the foundations of logic and philosophy". I remark that if it were about that then it would be very difficult for it also to be about geometric algebra, since geometric algebra doesn't have much to do with the foundations of logic and philosophy. In fact it doesn't seem to me that the linked post has much to do with the foundations of logic and philosophy either.
It invites us to consider a straight line on a piece of paper, and then a construction where a differently-coloured piece of paper is overlaid on the first one with an edge along that straight line, and then one where instead the two pieces of paper are the same colour. It then starts talking about "an object which we compare with itself it as it is represented on each side of our line", and -- to my mind, though of course it may be that I'm just not clever enough to understand the author's point(s) -- devolves further and further into word salad from there. "If our 0 width line is a number line and the parallel postulate is a statement about Cauchy completeness, what do our extended logics construct as number-like objects?" It means nothing to say that some specific geometrical line "is a number line"; the parallel postulate is not really a statement about Cauchy completeness, and the usual constructions of geometries where it fails aren't ones where anything strange is going on with the real numbers; the situations involving pieces of paper etc. are not "extended logics" in any useful sense I can see.
Anyway: none of that makes any reference to vector spaces (other than, implicitly, one particular vector space of dimension two), nor to Clifford algebras (even implicitly), nor to any sort of "algebra" construction (in the sense of a vector space with a multiplication operation defined on it). The only overlap between it and "geometric algebra" is that both have something to do with geometry. And even that seems kinda dubious, since the author's intention with the linked post is apparently to say something about "the foundations of logic" and that's taking things in an entirely different direction from anything to do with geometric algebra.
(I know you it doesn't let you, unless you have an alt account.)
I must say that I am a little surprised that this is something you have not come across before and I take it you are surprised as well. Apparently foundations are not taught in every school.
When we are talking about foundations we are usually concerned with something that prompted Euclid to write a very long time ago. In his Book 2 he defines geometric algebra for the first time but it is his parallel postulate which gets the most scrutiny because it is an outlier amongst his other claims. Elements is widely regarded as instrumental for logic and "natural philosophy" which is today called science.
I'm the author of that paper construction I linked to. The thought experiment I am offering was something I invented 30 years ago and I have found much use for it in my studies of ontology. It is very simple and directly let me discover logics that do not rely on the law of the excluded middle and a rudimentary quantum logic. However its greatest strength is how it supports your intuition and gives legitimacy to your imagination.
Cauchy sequences are what shows us that we have a way to actually reach every Real number, with no gaps, and this is what lets us talk about the number line as a mathematical object. Without this we don't have a foundation for differentiation. Perhaps now you can see why we might want to construct alternative number-like objects?
If your thought experiment involving pieces of paper has helped you to think of useful or interesting things that you wouldn't otherwise have thought of, that's great. It hasn't so far done anything of the kind for me. The fault could of course be mine.
What I said was not (as you claimed in another comment in this thread) that I don't see how constructivism is relevant -- though in fact I don't think constructivism is relevant -- but that I don't see how your thought experiment about pieces of paper is relevant.
I regret that it wasn't immediately apparent to me that your thought experiment was about constructivism. Now that you say it is, I can kinda see how it kinda relates, but for me thinking about your pieces of paper doesn't conjure up any useful insights about constructivism, and if I didn't already know about constructivism I don't think it would particularly point me in that direction. Again, if it works that way for you, good for you, but I think that when you present it to other people you are forgetting that the others (who unlike you haven't been thinking about this picture for 30 years) don't have the same associations with your imaginary pieces of paper that you have.
I have no problem with the idea that one might want to construct alternative number-like objects. (And yes, I know what Cauchy sequences are and what one does with them. And no, they really don't have very much to do with the parallel postulate. There's an analogy between the parallel postulate and, say, the LEM, but that's all.) But (1) again, this has nothing much to do with Eric Lengyel's critique of how geometric algebra is presented -- when Lengyel talks about "foundations" he isn't talking about going down to the level of set theory, logic, HOTT, or whatever, he's talking about getting the basics of geometric algebra right when one already has notions like "real numbers", "vector space", etc., in hand -- and (2) your thought experiment happens not to do anything for me to clarify, inspire, etc., ideas about alternative number-like objects. (Also, it seems to me that if you want to use that thought experiment to say something about number lines then you want your 0-width lines, half-planes, etc., to lie across the number line, not along it.)
It is simply not true that book 2 of Euclid "defines geometric algebra for the first time" or indeed defines it at all. Euclid is doing geometry but is not doing geometric algebra, which is something more specific.
"What can we do with the black-on-white paper construction to extract information from the system?"
I will need to leave this discussion with you here. I only mean to hold open the door for you and to give you two more variations of the tools that were used to define this long tradition of rational inquiry.
No need to assume that downvotes convey anger at all. They are given for lots of reasons, including style, rudeness, off-topic comments, irrelevance, incorrect or wrong comments, assumptions, negativity, etc. Sometimes downvotes (and upvotes) are nothing more than prioritizing. All that makes your edit and last sentence especially prone to downvotes, it’s multiple negative incorrect assumptions, and goes against HN guidelines.
As you're fond of the guidelines, you have probably also read the bits where they ask you to avoid interrogating, to not to go on about votes and not to sneer, especially at your fellow co-commentators.
I don't have any assumptions about you, it's just stuff you should avoid in comments because it tends to trash the conversation and, like the guidelines point out, isn't interesting.
